This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384988 #46 Jul 26 2025 10:57:07
%S A384988 0,1,10,55,250,1051,4270,17095,68050,270451,1075030,4276735,17030650,
%T A384988 67881451,270777790,1080817975,4316294050,17244046051,68912400550,
%U A384988 275457464815,1101251874250,4403270396251,17607863991310,70415790601255,281616141147250,1126323450484051
%N A384988 a(n) = Stirling2(n,2)^2 + Stirling2(n,3).
%C A384988 Also, one third of the number of proper vertex colorings of the n-complete tripartite graph using exactly 5 interchangeable colors.
%C A384988 The complete 3-partite graph K(n,n,n) has 3n vertices partitioned into three sets of size n each, with edges between every pair of vertices from different sets. 3*a(n) = 0 for n < 2 because we need at least 2 vertices per partition to create 5 nonempty independent sets.
%H A384988 Vincenzo Librandi, <a href="/A384988/b384988.txt">Table of n, a(n) for n = 1..500</a>
%H A384988 Richard P. Stanley, <a href="https://math.mit.edu/~rstan/ec/">Enumerative Combinatorics</a>, Cambridge University Press.
%H A384988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteMultipartiteGraph.html">Complete Multipartite Graph</a>.
%H A384988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling Number of the Second Kind</a>.
%H A384988 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F A384988 3*a(n) = 2^(2*n - 2) + (1/2)*3^(n - 1) - 3*2^(n - 1) + 3/2 for n >= 1.
%F A384988 G.f.: 1/(4*(1 - 4*x)) + 1/(6*(1 - 3*x)) - 3/(2*(1 - 2*x)) + 3/(2*(1 - x)).
%F A384988 a(n) = A385432(n, 5) / 3 = A060867(n-1) + A000392(n).
%F A384988 From _Stefano Spezia_, Jun 14 2025: (Start)
%F A384988 a(n) = (6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4.
%F A384988 E.g.f.: (exp(x) - 1)^2*(3*exp(2*x) + 8*exp(x) - 5)/12. (End)
%F A384988 a(n) = A000453(n+2) -10*A000453(n). - _R. J. Mathar_, Jul 20 2025
%e A384988 3*a(2) = 3 because K(2,2,2) can be partitioned into 5 nonempty independent sets in exactly 3 ways.
%t A384988 Table[(StirlingS2[n, 3] + StirlingS2[n, 2]^2), {n, 1, 20}]
%o A384988 (Magma) [(6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4: n in [1..30]]; // _Vincenzo Librandi_, Jul 24 2025
%Y A384988 Cf. A000392, A000453, A060867, A385432.
%K A384988 nonn,easy
%O A384988 1,3
%A A384988 _Julian Allagan_, Jun 14 2025